The South African may not have got his name on the scoresheet but was instrumental in a lively first-half display which saw tries from Paul Marshall, Darren Cave, Louis Ludik and Jacob Stockdale, and while Ulster flagged somewhat in the second period, the five-point victory keeps the pressure on Glasgow Warriors and Scarlets for a play-off place.
It was a momentous night not only for marquee signing Coetzee, but also for home-grown talent Peter Nelson, the utility back making his first appearance in 15 months after a long rehabilitation from an ankle injury sustained in the self-same fixture in the 2015/16 campaign.
Starting at out-half, Nelson was joined in the backline by centre Stuart Olding, back in action for the first time since October, while Ludik started at full-back, Stockdale and Charles Piutau on the wings, Cave at 12, and Marshall at scrum-half.
Up front the absence of both Rory Best and Rob Herring gave John Andrew the chance to impress at hooker on his third senior start, while the remainder of an unfamiliar front five comprised props Andrew Warwick and Ricky Lutton and a second row of Kieran Treadwell and Alan O'Connor. Robbie Diack and Chris Henry lined up alongside Coetzee in an all-international back row.
Edinburgh were on the scoresheet within 120 seconds, Number Eight - and club top scorer this term with four at start of play - Marcus Bradbury finishing off a fluent move by the left-hand corner flag.
As Ulster settled into the game two close calls on the right wing saw both Stockdale and Cave bundled into touch five metres from home in quick succession, before neat interplay in the next attack between Olding and Marshall got the scrum-half over on the 10-minute mark, Nelson adding the extras.
Channelling the ball through Coetzee as often as possible, Ulster repeatedly made good ground through the middle, and when the Springbok pinched possession at the Edinburgh put-in, a muscular run to the line from Stockdale - with several Scots hanging off his frame but unable to bring him down - paved the way for Cave's third try of the season.
After a lucky escape at the other end as the Edinburgh half-backs contrived to knock on five metres from the line after eluding last man Nelson, Ludik consolidated the Ulster lead with the third try, engineered by yet another slinky run from Piutau which tied the visiting defenders up in knots.
The bonus point was sewn up just after the half-hour mark, as Diack broke clear down the right flank to feed Stockdale, whose impressive turn of pace eliminated his chasers to allow the grounding underneath the posts.
Poor Ulster reflexes at the restart let hooker Stuart McInally in for a soft score on 35 minutes, but outstanding wingplay from Stockdale and Piutau again - an aerial take from the youngster and another pacy break from the Kiwi - saw Ulster unfortunate to finish the half without a fifth try.
Half-Time Score Ulster 24 Edinburgh 10
The half-time break did little to quell Ulster's appetite, and with the brute force of Stuart McCloskey and Wiehahn Herbst now in the mix in place of Cave and Lutton, Edinburgh struggled even to get hands on the ball for a good 10 minutes.
Piutau's game ended on 50 minutes as he limped off after an awkward landing, replaced by Tommy Bowe, and as Edinburgh edged their way back into the game Damien Hoyland snuck in for their third score in the right hand corner just before the hour.
Coetzee made way for Sean Reidy to earn his fiftieth cap just after the concession, and after a protracted pause for treatment to Fraser McKenzie, injured in a collision with Bowe, Sam Hidalgo-Clyne's penalty brought the visitors to within six points of parity.
Ulster did enough to contain Edinburgh, who now scented blood, in the middle of the park as the clock ran down through its last 10 minutes, and while their blank second half scorecard will be a source of worry and frustration for Les Kiss, the first-half offensive display and maximum points haul will be positives to take into next Saturday's clash in Belfast with fellow contenders Glasgow Warriors.
Full-Time Score Ulster 24 Edinburgh 18
Ulster (15 - 9) Louis Ludik; Jacob Stockdale, Darren Cave, Stuart Olding, Charles Piutau; Peter Nelson, Paul Marshall
(1 - 8) Andrew Warwick, John Andrew, Ricky Lutton, Kieran Treadwell, Alan O'Connor, Robbie Diack, Chris Henry (c), Marcell Coetzee
Replacements (16 - 23) Jonny Murphy, Callum Black, Wiehahn Herbst, Pete Browne, Sean Reidy, David Shanahan, Stuart McCloskey, Tommy Bowe
Edinburgh (15 - 9) Blair Kinghorn; Damien Hoyland, Chris Dean, Phil Burleigh, Tom Brown; Jason Tovey, Sean Kennedy
(1 - 8) Jack Cosgrove, Stuart McInally (c), Murray McCallum, Fraser McKenzie, Ben Toolis, Viliame Mata, Jamie Ritchie, Magnus Bradbury
Replacements (16 - 23) George Turner, Derrick Appiah, Nick Beavon, Lewis Carmichael, Viliami Fihaki, Sam Hidalgo-Clyne, Mike Allen, Rory Scholes
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